The reason for having a SAHP with school age children flips from financial to logistical...
First year of school is often messy with absences - DC1 is in reception and has had several vomiting bugs and chicken pox in 1 and a half terms - having spent all of last year doing mornings in the same classroom. She has shared many of these presents with DH and her younger sibling (I seem to be more resistant so far). She is also quite tough with colds - has been fine with colds that have floored DH, DC2 and me... trying to sustain 2 jobs through the first term would have been impossible without a nanny!
Summer born DC can't do most holiday clubs as childcare for their first year - they have to be 5 to have more than 4 hours - again making childcare logistics complicated - they have to go to nursery based holiday clubs.
Also schools away from standard commuter areas have spotty wrap around care - my DC's school has after school club to 6pm 2 days, 5pm 1 day and none (so collect by 3.10) the other 2. Getting a job that fits that pattern would be hard even without a commute, nearly impossible with a commute.
If the working parent is willing/able to flex their hours to allow the other parent some freedom in their job hunt it could work, but without that option (shift work, lack of flexibility, frequent travel) there's definitely a need to balance the logistical headaches with the desire to work.
I'm currently a SAHP, DH has an inflexible job and we are moving in the hope of making the logistics of getting back to work possible!